Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

NUMBER OF WEAK GALOIS-WEIERSTRASS POINTS WITH WEIERSTRASS SEMIGROUPS GENERATED BY TWO ELEMENTS

  • Komeda, Jiryo (Department of Mathematics Center for Basic Education and Integrated Learning Kanagawa Institute of Technology) ;
  • Takahashi, Takeshi (Education Center for Engineering and Technology Faculty of Engineering Niigata University)
  • Received : 2018.11.01
  • Accepted : 2019.07.25
  • Published : 2019.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let C be a nonsingular projective curve of genus ${\geq}2$ over an algebraically closed field of characteristic 0. For a point P in C, the Weierstrass semigroup H(P) is defined as the set of non-negative integers n for which there exists a rational function f on C such that the order of the pole of f at P is equal to n, and f is regular away from P. A point P in C is referred to as a weak Galois-Weierstrass point if P is a Weierstrass point and there exists a Galois morphism ${\varphi}:C{\rightarrow}{\mathbb{p}}^1$ such that P is a total ramification point of ${\varphi}$. In this paper, we investigate the number of weak Galois-Weierstrass points of which the Weierstrass semigroups are generated by two positive integers.

Keywords

References

  1. W. Barth, C. Peters, and A. van de Ven, Compact Complex Surfaces, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-57739-0
  2. W. Castryck and F. Cools, Linear pencils encoded in the Newton polygon, Int. Math. Res. Not. IMRN 2017 (2017), no. 10, 2998-3049. https://doi.org/10.1093/imrn/rnw082
  3. H. C. Chang, On plane algebraic curves, Chinese J. Math. 6 (1978), no. 2, 185-189.
  4. M. Coppens, The uniqueness of Weierstrass points with semigroup (a; b) and related semigroups, Abh. Math. Semin. Univ. Hambg., published online: 15 February 2019. https://doi.org/10.1007/s12188-019-00201-y
  5. S. Fukasawa, On the number of Galois points for a plane curve in positive characteristic. III, Geom. Dedicata 146 (2010), 9-20. https://doi.org/10.1007/s10711-009-9422-x
  6. S. Fukasawa and H. Yoshihara, List of problems, available at http://hyoshihara.web.fc2.com/openquestion.html.
  7. J. Komeda and T. Takahashi, Relating Galois points to weak Galois Weierstrass points through double coverings of curves, J. Korean Math. Soc. 54 (2017), no. 1, 69-86. https://doi.org/10.4134/JKMS.j150593
  8. J. Komeda and T. Takahashi, Galois Weierstrass points whose Weierstrass semigroups are generated by two elements, arXiv:1703.09416 (2017).
  9. K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283-294. https://doi.org/10.1006/jabr.1999.8173
  10. I. Morrison and H. Pinkham, Galois Weierstrass points and Hurwitz characters, Ann. of Math. (2) 124 (1986), no. 3, 591-625. https://doi.org/10.2307/2007094
  11. H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340-355. https://doi.org/10.1006/jabr.2000.8675